Local properties of non-negative solutions to some doubly non-linear degenerate parabolic equations
Simona Fornaro Ugo Gianazza
Discrete & Continuous Dynamical Systems - A 2010, 26(2): 481-492 doi: 10.3934/dcds.2010.26.481
In the present paper we study the local behavior of non-negative weak solutions of a wide class of doubly non linear degenerate parabolic equations. We show, in particular, some lower pointwise estimates of such solutions in terms of suitable sub-potentials (dictated by the structure of the equation) and an alternative form of the Harnack inequality.
keywords: degenerate parabolic equations Doubly non-linear local properties.
Very weak solutions of singular porous medium equations with measure data
Verena Bögelein Frank Duzaar Ugo Gianazza
Communications on Pure & Applied Analysis 2015, 14(1): 23-49 doi: 10.3934/cpaa.2015.14.23
We consider non-homogeneous, singular ($ 0 < m < 1 $) porous medium type equations with a non-negative Radon-measure $\mu$ having finite total mass $\mu(E_T)$ on the right-hand side. We deal with a Cauchy-Dirichlet problem for these type of equations, with homogeneous boundary conditions on the parabolic boundary of the domain $E_T$, and we establish the existence of a solution in the sense of distributions. Finally, we show that the constructed solution satisfies linear pointwise estimates via linear Riesz potentials.
keywords: existence boundedness. Singular porous medium equations very weak solutions Riesz potential
$1$-dimensional Harnack estimates
Fatma Gamze Düzgün Ugo Gianazza Vincenzo Vespri
Discrete & Continuous Dynamical Systems - S 2016, 9(3): 675-685 doi: 10.3934/dcdss.2016021
Let $u$ be a non-negative super-solution to a $1$-dimensional singular parabolic equation of $p$-Laplacian type ($1< p <2$). If $u$ is bounded below on a time-segment $\{y\}\times(0,T]$ by a positive number $M$, then it has a power-like decay of order $\frac p{2-p}$ with respect to the space variable $x$ in $\mathbb R\times[T/2,T]$. This fact, stated quantitatively in Proposition 1.2, is a ``sidewise spreading of positivity'' of solutions to such singular equations, and can be considered as a form of Harnack inequality. The proof of such an effect is based on geometrical ideas.
keywords: Singular diffusion equations expansion of positivity. $p$-laplacian
On the local behavior of non-negative solutions to a logarithmically singular equation
Emmanuele DiBenedetto Ugo Gianazza Naian Liao
Discrete & Continuous Dynamical Systems - B 2012, 17(6): 1841-1858 doi: 10.3934/dcdsb.2012.17.1841
The local positivity of solutions to logarithmically singular diffusion equations is investigated in some open space-time domain $E\times(0,T]$. It is shown that if at some time level $t_o\in(0,T]$ and some point $x_o\in E$ the solution $u(\cdot,t_o)$ is not identically zero in a neighborhood of $x_o$, in a measure-theoretical sense, then it is strictly positive in a neighborhood of $(x_o, t_o)$. The precise form of this statement is by an intrinsic Harnack-type inequality, which also determines the size of such a neighborhood.
keywords: logarithmic diffusion Singular parabolic equations expansion of positivity Harnack-type estimates.

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